Generic singular continuous spectrum for ergodic Schr\"odinger operators

TL;DR

This paper demonstrates that for a broad class of ergodic Schrödinger operators, the spectrum is generically singular continuous, with no absolutely continuous part, using approximation techniques and Kotani Theory.

Contribution

It establishes that for generic continuous potentials, the spectrum of ergodic Schrödinger operators lacks absolutely continuous components, extending understanding of spectral types in these systems.

Findings

Spectrum is generically singular continuous for ergodic Schrödinger operators.

No absolutely continuous spectrum exists for generic continuous potentials.

Proof utilizes approximation by discontinuous potentials and Kotani Theory.

Abstract

We consider Schr\"odinger operators with ergodic potential $V_\omega(n)=f(T^n(\omega))$, $n \in \Z$, $\omega \in \Omega$, where $T:\Omega \to \Omega$ is a non-periodic homeomorphism. We show that for generic $f \in C(\Omega)$, the spectrum has no absolutely continuous component. The proof is based on approximation by discontinuous potentials which can be treated via Kotani Theory.